Embedded newform 21.33.h.b.11.1

• Label: 21.33.h.b.11.1
• Level: $21$
• Weight: $33$
• Character: 21.11
• Analytic conductor: $136.220$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 33 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(136.219975799\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	21.11
Dual form			21.33.h.b.2.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-111997. + 64661.2i) q^{2} +(4.13267e7 - 1.20467e7i) q^{3} +(6.21466e9 - 1.07641e10i) q^{4} +(1.71366e11 - 9.89381e10i) q^{5} +(-3.84949e12 + 4.02143e12i) q^{6} +(2.06412e13 - 2.60455e13i) q^{7} +1.05196e15i q^{8} +(1.56277e15 - 9.95702e14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	164 * q + 43046720 * q^3 + 176093659134 * q^4 + 3915256954876 * q^6 - 41868332499126 * q^7 + 1232720974221986 * q^9

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−111997.	+	64661.2i	−1.70893	+	0.986652i	−0.773044	+	0.634352i	\(0.781267\pi\)
							−0.935887	+	0.352300i	\(0.885400\pi\)
\(3\)	4.13267e7	−	1.20467e7i	0.960043	−	0.279852i
							\(5\)	1.71366e11	−	9.89381e10i	1.12306	−	0.648401i	0.180882	−	0.983505i	\(-0.442105\pi\)
0.942181	+	0.335104i	\(0.108771\pi\)
\(7\)	2.06412e13	−	2.60455e13i	0.621105	−	0.783727i
							\(11\)	3.51850e16	+	2.03141e16i	0.765729	+	0.442094i	0.831349	−	0.555751i	\(-0.187569\pi\)
−0.0656200	+	0.997845i	\(0.520903\pi\)
\(13\)	6.93583e17			1.04233			0.521165	−	0.853456i	\(-0.325498\pi\)
							0.521165	+	0.853456i	\(0.325498\pi\)
\(17\)	3.76714e18	+	2.17496e18i	0.0774157	+	0.0446960i	0.538208	−	0.842812i	\(-0.319101\pi\)
							−0.460792	+	0.887508i	\(0.652435\pi\)
\(19\)	−1.55399e20	−	2.69159e20i	−0.538754	−	0.933150i	−0.998971	−	0.0453433i	\(-0.985562\pi\)
							0.460217	−	0.887806i	\(-0.347771\pi\)
\(23\)	8.37655e21	−	4.83620e21i	1.36590	−	0.788604i	0.375500	−	0.926822i	\(-0.377471\pi\)
							0.990402	+	0.138218i	\(0.0441375\pi\)
\(29\)			1.11691e22i			0.0446324i	0.999751	+	0.0223162i	\(0.00710405\pi\)
							−0.999751	+	0.0223162i	\(0.992896\pi\)
\(31\)	1.03293e23	−	1.78908e23i	0.141998	−	0.245948i	−0.786251	−	0.617907i	\(-0.787981\pi\)
							0.928249	+	0.371960i	\(0.121314\pi\)
\(37\)	7.93567e24	+	1.37450e25i	0.643215	+	1.11408i	0.984711	+	0.174197i	\(0.0557331\pi\)
							−0.341496	+	0.939883i	\(0.610934\pi\)
\(41\)			3.16825e25i			0.496910i	0.968643	+	0.248455i	\(0.0799229\pi\)
							−0.968643	+	0.248455i	\(0.920077\pi\)
\(43\)	1.07349e26			0.785781			0.392891	−	0.919585i	\(-0.371475\pi\)
							0.392891	+	0.919585i	\(0.371475\pi\)
\(47\)	1.81600e26	−	1.04847e26i	0.320295	−	0.184922i	−0.331229	−	0.943550i	\(-0.607463\pi\)
							0.651524	+	0.758628i	\(0.274130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.b.11.1	yes	164
3.2	odd	2	inner	21.33.h.b.11.82	yes	164
7.2	even	3	inner	21.33.h.b.2.82	yes	164
21.2	odd	6	inner	21.33.h.b.2.1	✓	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.b.2.1	✓	164	21.2	odd	6	inner
21.33.h.b.2.82	yes	164	7.2	even	3	inner
21.33.h.b.11.1	yes	164	1.1	even	1	trivial
21.33.h.b.11.82	yes	164	3.2	odd	2	inner